Dominated Splitting from Constant Periodic Data and Global Rigidity of Anosov Automorphisms

We show that a \(\operatorname{GL}(d,\mathbb{R})\) cocycle over a hyperbolic system with constant periodic data has a dominated splitting whenever the periodic data indicates it should. This implies global periodic data rigidity of generic Anosov automorphisms of \(\mathbb{T}^{d}\). Further, our approach also works when the periodic data is narrow, that is, sufficiently close to constant. We can show global periodic data rigidity for certain non-linear Anosov diffeomorphisms in a neighborhood of an irreducible Anosov automorphism with simple spectrum.